Starter Problems

Strang Matrix Problem

In [3]:
N = 10
A = zeros(N,N)
for i in 1:N, j in 1:N
    abs(i-j)<=1 && (A[i,j]+=1)
    i==j && (A[i,j]-=3)
end
A
Out[3]:
10×10 Array{Float64,2}:
 -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0
  1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0
  0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0
  0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0
  0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0
  0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0
  0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0
  0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0
  0.0   0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0
  0.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0

Factorial Problem

In [1]:
function my_factorial(n)
    k = one(n)
    for i in 1:n
        k *= i
    end
    k
end

my_factorial(4)
my_factorial(30)
my_factorial(big(30))
Out[1]:
265252859812191058636308480000000

Binomial Problem

In [1]:
function binomial_rv(n, p)
    count = zero(n)
    U = rand(n)
    for i in 1:n
      U[i] < p && (count += 1)
    end
    count
end

bs = [binomial_rv(10, 0.5) for j in 1:10]
Out[1]:
10-element Array{Int64,1}:
 1
 5
 6
 6
 2
 4
 3
 5
 5
 8

Monte Carlo $\pi$ Problem

In [17]:
n = 10000000

count = 0
for i in 1:n
    u, v = 2rand(2)-1
    d = sqrt(u^2 + v^2)  # Distance from middle of square
    d < 1 && (count += 1)
end

area_estimate = count / n

print(area_estimate * 4)  # dividing by radius**2

3.1417656

Integration Problems

Timeseries Generation Problem

In [22]:
alphas = [0.0, 0.5, 0.98]
T = 200

series = []
labels = []

for alpha in alphas
    x = zeros(T + 1)
    x[1] = 0.0
    for t in 1:T
        x[t+1] = alpha * x[t] + randn()
    end
    push!(series, x)
    push!(labels, "alpha = $alpha")
end

plot(series, label=reshape(labels,1,length(labels)),lw=3)
Out[22]:
50 100 150 200 0 5 alpha = 0.0 alpha = 0.5 alpha = 0.98

Linear Regression Problem

In [2]:
#### Prepare Data

X = rand(1000, 3)               # feature matrix
a0 = rand(3)                    # ground truths
y = X * a0 + 0.1 * randn(1000);  # generate response

X2 = hcat(X,ones(1000))
println(X2\y)

using MultivariateStats
println(llsq(X,y))

using DataFrames, GLM
data = DataFrame(X1=X[:,1], X2=X[:,2], X3=X[:,3],Y=y)
OLS = lm(@formula(Y ~ X1 + X2 + X3), data)


X = rand(100);
y = 2X  + 0.1 * randn(100);

using Plots
b = X\y
println(b)
gr()
scatter(X,y)
Plots.abline!(b[1],0.0, lw=3) # Slope,Intercept

[0.474905,0.910809,0.298824,0.00130641]
[0.474905,0.910809,0.298824,0.00130641]
[1.9975]

WARNING: Method definition describe(AbstractArray) in module StatsBase at /home/crackauc/.julia/v0.5/StatsBase/src/scalarstats.jl:573 overwritten in module DataFrames at /home/crackauc/.julia/v0.5/DataFrames/src/abstractdataframe/abstractdataframe.jl:407.
Out[2]:
0.0 0.5 1.0 0 1 2 y1 y2

Logistic Equation Problem

In [9]:
r = 2.9:.001:4; numAttract = 100
steady = ones(length(r),1)*.25
for i=1:400 ## Get to steady state
  steady .= r.*steady.*(1-steady)
end
x = zeros(length(steady),numAttract)
x[:,1] = steady
@inbounds for i=2:numAttract ## Grab values at the attractor
  x[:,i] = r.*x[:,i-1].*(1-x[:,i-1])
end
using Plots; gr()
plot(collect(r),x,seriestype=:scatter,markersize=.002,legend=false,color=:black)
Out[9]:
3.0 3.5 4.0 0.0 0.5 1.0